Computing rates umbrella sampling

Through a procedure such as umbrella sampling we can calculate the correlation function C(t) for a particular time t. For a determination of the reaction rate constants, however, we need the derivative of C(t). Of course, the time derivative of C(t) could be determined by calculating C(t) at different path lengths t and taking the derivative numerically. Fortunately, such a computationally expensive procedure is not necessary. One can derive expressions for the reaction rate constant that... 

Umbrella sampling for ensembles of trajectories can be carried out in close analogy to the procedure described above. For the purpose of computing rate constants (in this case the rate of conformational transitions), we focus on computing the reversible work to confine trajectories endpoints to state B, given that these trajectories begin in state A. P (A, t) is defined to be the distribution of A at the endpoints of trajectories initiated in A,... 

In order to study crystal nucleation at moderate supersaturation, we exploit the fact that the crystallization rate is determined by the product of a static term, namely the probability for the formation of a critical nucleus Pc, and a kinetic factor F that describes the rate at which such nuclei grow. We use umbrella sampling to compute Pc and kinetic Monte Carlo simulations to compute F. The computed nucleation rates can be directly compared to experimental data. 

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